Numer. Math. 74: 1–33 (1996) Numerische Mathematik c © Springer-Verlag 1996

Summary. For the high-order numerical approximation of hyperbolic systems of conservation laws, we propose to use as a building principle an entropy di-minishing criterion instead of the familiar total variation diminishing criterion introduced by Harten for scalar equations. Based on this new criterion, we derive entropy diminishing projections that ensure, both, the second order of accuracy and all of the classical discrete entropy inequalities. The resulting scheme is a nonlinear version of the classical Van Leer’s MUSCL scheme. Strong conver-gence of this second order, entropy satisfying scheme is proved for systems of two equations. Numerical tests demonstrate the interest of our theory. Mathematics Subject Classification (1991): 65M05, 65M12, 35L65 1